Question

A chain letter works as follows: One person sends a copy of the letter to five friends, each of whom sends a copy to five friends, each of whom sends a copy to five friends, and so forth. How many people will have received copies of the let- ter after the twentieth repetition of this process, assuming no person receives more than one copy?

Answer 1

The number of people that received copies of the letter at the twentieth stage is 9.537 × 10¹³ .

Explanation:

Using the discrete model,

a_k = r a_(k-1) for all integers k ≥ 1 and a₀ = a

then,

aₙ = a rⁿ for all integers n ≥ 0

Let a_k be equal to the number of people who receive a copy of the chain letter at a stage k.

Initially, one person has the chain letter (which the person will send to five other people at stage 1). Thus,

a = a₀ = 1

The people who received he chain letter at stage (k - 1), will send a letter to five people at stage k and thus per person at stage (k - 1), five people will receive the letter. Therefore,

a_k = 5 a_(k - 1)

Thus,

aₙ = a rⁿ = 1 · 5ⁿ = 5ⁿ

The number of people that received copies of the letter at the twentieth stage is

a₂₀ = (5)²⁰ = 9.537 × 10¹³ copies